7. GALAXY TYPES AND ROTATION CHARACTERISTICS

There is a marked similarity of form, but not of amplitude, of
disk and halo rotation curves
for galaxies with different morphologies from Sa to Sc
(Rubin et al. 1985).
Thus the form of the gravitational potential in the disk and halo
is not strongly dependent on the form of the optical luminosity
distribution. Some moderate correlation is found between total luminosity
and rotation velocity amplitude.
Also, less luminous galaxies tend to show increasing outer rotation curve,
while most massive galaxies have slightly declining rotation in the
outmost part
(Persic et al. 1996).
On the other hand, form of central rotation curves depend on the total
mass and galaxy types
(Sofue et al. 1999):
Massive galaxies of Sa and Sb types show a steeper rise and
higher central velocities within a few hundred pc of the nucleus
compared to less massive Sc galaxies and dwarfs.
Dwarf galaxies generally show a gentle central rise.

The maximum rotation velocities for Sa galaxies are higher than those of Sb
and Sc galaxies with equivalent optical luminosities. Median values of
Vmax decreases from 300 to 220 to 175 km
s-1 for the Sa, Sb, and Sc types, respectively
(Rubin et al. 1985).

Sb galaxies have rotation curves with slightly lower
values of the maximum velocity than Sa
(Rubin et al. 1982).
The steep central rise at 100-200 pc is often associated
with a velocity peak at radii r ~ 100 - 300 pc
(Sofue et al. 1999a).
The rotation velocity then declines to a minimum at r ~ 1 kpc,
and is followed by a gradual rise to a broad maximum at
r ~ 2 - 7 kpc, arising from the disk potential. The disk rotation
curve has superposed amplitude fluctuations of tens of km s-1
due to spiral arms or velocity ripples.
The outermost parts are usually flat, due to the massive dark halo.
Some Sb galaxies show a slight outer decline, often no larger than the inner
undulations
(Honma & Sofue 1997a,
b).

The rotation curve of the Milky Way Galaxy, a typical Sb galaxy,
is shown in Fig. 2
(Clemens 1985;
Blitz 1979;
Brand and Blitz 1993;
Fich et al. 1989;
Merrifield 1992;
Honma and Sofue 1995).
The rotation curve of Sb galaxies, including the Milky Way,
can be described as having:
(a) a high-density core, including the massive black hole, which causes a
non-zero velocity very close to the center;
(b) a steep rise within the central 100 pc;
(c) a maximum at radius of a few hundred pc, followed by a decline to a
minimum at 1 to 2 kpc; then,
(d) a gradual rise from to the disk maximum at 6 kpc; and
(e) a nearly flat outer rotation curve.

Sc galaxies have lower maximum velocities than Sa and Sb
(Rubin et al. 1980,
1985),
ranging from
100 to ~ 200 km s-1.
Massive Sc galaxies show a steep nuclear rise similar to Sb's.
However, less-massive Sc galaxies have a more gentle rise.
They also have a flat rotation to their outer edges.
Low-surface brightness Sc galaxies have a gentle central
rise with monotonically increasing rotation velocity toward the edge,
similar to dwarf galaxies
(Bosma et al. 1988).

Large-scale rotation properties of SBb and SBc galaxies are generally
similar to those of non-barred galaxies of Sb and Sc types.
However, the study of their kinematics is more
complicated than for non-barred spirals, because their gas tracers
are less uniformly distributed
(Bosma 1981a,
1996),
and their iso-velocity contours are skewed in the direction toward the bar
(H,
Peterson et al. 1978;
HI,
Sancisi et al. 1979;
stellar absorption lines,
Kormendy 1983).
CO-line mapping and spectroscopy reveal high concentration of
molecular gas in shocked lanes along a bar superposed by significant
non-circular motions
(Handa et al. 1990;
Sakamoto et al. 1999).

This large velocity variation arises from the barred
potential of several kpc length.
Simulations of PV diagrams for edge-on barred galaxies
show many tens of km s-1 fluctuations,
superposed on the usual flat rotation curve
(Athanassoula and Bureau
1999;
Bureau and Athanassoula
1999;
Weiner & Sellwood
1999).
However, distinguishing the existence of a bar and quantifying it
are not uniquely done from such limited edge-on information.
For more quantitative results, two-dimensional velocity analyses are
necessary
(Wozniak & Pfenniger
1997).
In these models, barred spirals contain up to 30% counterrotating stars;
the orbits are almost circular and perpendicular to the bar.
Pattern speeds for the bar have been determined from absorption line spectra
(Buta et al. 1996;
Gerssen 2000
and references therein).

Until the last decade, observations of rotational kinematics were
restricted to spirals with average or high surface brightness.
Only within the past decade have low surface brightness (LSB)
galaxies been found in great numbers
(Schombert & Bothun
1988;
Schombert et al. 1992);
many are spirals. Their kinematics were first studied by
de Blok et al. (1996)
with HI, who found slowly rising curves which often
continued rising to their last measured point.
However, many of the galaxies are small in angular extent,
so observations are subject to beam smearing.
Recent optical rotation curves
(Swaters 1999,
2001;
Swaters et al. 2000;
de Blok et al. 2001)
reveal a steeper rise for some,
but not all, of the galaxies studied previously at 21-cm.
It is not now clear if LSB galaxies are as dominated by dark matter
as they were previously thought to be; the mass models
have considerable uncertainties.

Swaters (1999)
derived rotation curves from velocity fields obtained with the
Westerbork Synthesis Radio Telescope for 60 late-type dwarf galaxies of low
luminosity. By an interactive analysis, he obtained rotation curves which
are corrected for a large part of the beam smearing. Most of the rotation
curve shapes are similar to those of more luminous spirals; at the lowest
luminosities, there is more variation in shape.
Dwarfs with higher central light concentrations have more steeply
rising rotation curves, and a similar dependence is found
for disk rotation curves of spirals (Fig. 5).
For dwarf galaxies dominated
by dark matter, as for LSB (and also HSB) spirals, the contributions of the
stellar and dark matter components to the total mass cannot be unambiguously
derived. More high quality observations and less ambiguous mass
deconvolutions, perhaps more physics, will be required to settle questions
concerning the dark matter fraction as a function of mass and/or luminosity.

Figure 5. The rotation curve slope between
one and two
scale lengths, plotted against the central concentration of light,
µR, which represents the
difference between the observed central surface brightness and
the extrapolated surface brightness of the exponential disk.
For pure disk systems,
µR = 0.
Filled and open circles are dwarfs (high and low accuracy), and
triangles are spirals. [Courtesy of
R. Swaters (1999)].

The LMC is a dwarf galaxy showing irregular optical
morphology, with the enormous starforming region, 30 Dor, located
significantly displaced from the optical bar and HI disk center.
High-resolution HI kinematics of the Large Magellanic Cloud,
Kim et al. (1998;
see Westerlund 1999
review) indicate, however, a regular
rotation around the kinematical center, which is displaced
1.2 kpc from the center of the optical bar as well as from the center of
starforming activity (Fig. 6).
The rotation curve has a steep central rise, followed by a flat rotation
with a gradual rise toward the edge. This implies that the LMC has a compact bulge
(but not visible on photographs),
an exponential disk, and a massive halo.
This dynamical bulge is 1.2 kpc away from the center of the stellar bar,
and is not associated with an optical counterpart.
The "dark bulge" has a large fraction of dark matter,
with an anomalously high mass-to-luminosity (M / L) ratio
(Sofue 1999).
In contrast, the stellar bar has a smaller M / L ratio
compared to that of the surrounding regions.

Figure 6. The HI velocity field of the LMC superposed on an
H image,
and a position-velocity diagram across the kinematical major axis
(Kim et al. 1998:
Courtesy of S. Kim).
The ellipse indicates the position of the optical bar.
The thick line in the PV diagram traces the rotation curve, corrected
for the inclination angle of 33°.

The interacting galaxy NGC 5194 (M51) shows a very peculiar rotation
curve, which declines more rapidly than Keplerian at
R ~ 8 - 12 kpc.
This may be due to inclination varying with the radius, e.g. warping.
Because the galaxy is viewed nearly face-on (i = 20°), a
slight warp causes a large error in deriving the rotation velocity.
If the galaxy's outer disk at 12 kpc has an inclination as small as
i ~ 10°, such an apparently steep velocity decrease would be
observed even for a flat rotation.

When galaxies gravitationally interact, they tidally distort each other, and
produce the pathological specimens that had until recently defied
classification. In an innovative paper,
Toomre (1977; see also
Toomre & Toomre 1972;
Holmberg 1941)
arranged eleven known distorted galaxies "in rough order of completeness of
the imagined mergers" starting with the Antennae (NGC 4038/39) and
ending with
NGC 7252. Observers rapidly took up this challenge, and
Schweizer (1982)
showed that NGC 7252 is a late-stage merger, in which the
central gas
disks of the two original spirals still have separate identities.

There is now an extensive literature both observational and computational
(Schweizer 1998
and references therein;
Barnes & Hernquist
1992,
1996;
Hibbard et al. 2000)
that make it possible to put limits on the initial
masses, the gas quantities, the time since the initial encounter, and the
evolutionary history of the merger remnant. Equally remarkable, tidal tails
can be used as probes of dark matter halos
(Dubinski et al. 1999).

The starburst dwarf galaxy NGC 3034 (M82)
shows an exceptionally peculiar rotation property
(Burbidge et al. 1964;
Sofue et al. 1992).
It has a normal steep nuclear rise and rotation velocities which have a
Keplerian decline beyond the nuclear peak. This may arise from a tidal
truncation of the disk and/or halo by an encounter with M81
(Sofue 1998).

A major surprise comes from the study of the polar ring galaxy NGC 4650A.
Arnaboldi et al. (1997;
see also
van Gorkom et al. 1987)
discovered an
extended HI disk coplanar with the ring, which twists from almost
edge-on to more face-on at large radii. The K-band optical features and the
HI velocities can be fit simultaneously with a model in which spiral
arms are
present in this polar disk. Hence the polar ring is a very massive disk.
This result strengthens previous suggestions that polar ring galaxies are
related to spirals
(Arnaboldi et al. 1995;
Combes & Arnaboldi
1996).